{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Further Hypothesis Testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "── Attaching packages ─────────────────────────────────────── tidyverse 1.3.0 ──\n",
      "✔ ggplot2 3.3.0     ✔ purrr   0.3.4\n",
      "✔ tibble  3.0.1     ✔ dplyr   0.8.5\n",
      "✔ tidyr   1.1.0     ✔ stringr 1.4.0\n",
      "✔ readr   1.3.1     ✔ forcats 0.4.0\n",
      "── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──\n",
      "✖ dplyr::filter() masks stats::filter()\n",
      "✖ dplyr::lag()    masks stats::lag()\n"
     ]
    }
   ],
   "source": [
    "# Select this cell and type Ctrl-Enter to execute the code below.\n",
    "\n",
    "library(tidyverse)\n",
    "\n",
    "set_plot_dimensions <- function(width_choice, height_choice) {\n",
    "    options(repr.plot.width=width_choice, repr.plot.height=height_choice)\n",
    "}\n",
    "\n",
    "cbPal <- c(\"#E69F00\", \"#56B4E9\", \"#009E73\", \"#F0E442\", \"#CC79A7\", \"#0072B2\", \"#D55E00\")\n",
    "\n",
    "set_plot_dimensions(5, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# You should see \"Attaching packages\" and some ticks by the packages loaded.\n",
    "# The \"Conflicts\" aren't a problem.\n",
    "\n",
    "# Other problems loading the library? Try running this cell.\n",
    "\n",
    "install.packages('tidyverse')\n",
    "\n",
    "library(tidyverse)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4 - Comparing means of more than two groups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Parsed with column specification:\n",
      "cols(\n",
      "  temperature = col_double(),\n",
      "  luminosity = col_double(),\n",
      "  radius = col_double(),\n",
      "  spectral_class = col_character(),\n",
      "  type = col_double()\n",
      ")\n"
     ]
    }
   ],
   "source": [
    "# Run this cell to load the data.\n",
    "\n",
    "data <- read_csv(\"stars.csv\")\n",
    "\n",
    "type_key <- c('Brown Dwarf', 'Red Dwarf', 'White Dwarf', 'Main Sequence', 'Supergiant','Hypergiant')\n",
    "spectral_classes <- c('O','B','A','F','G','K','M')\n",
    "\n",
    "data$type <- factor(data$type)\n",
    "data$spectral_class <- factor(data$spectral_class, levels=spectral_classes)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "During the course of your investigations for Dr Howe, you have noticed that the distributions of the dwarf stars' luminosities (types 0,1 and 2) are also overlapping."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wj9I0I99tuyZYupfpWKG/W4RpXcMyu4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\nRK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data %>%\n",
    "    filter(type %in% c(0,1,2)) %>%\n",
    "    ggplot(aes(x=log(luminosity), fill=type)) + \n",
    "        scale_fill_manual(values=cbPal[c(1,2,3)]) +\n",
    "        geom_histogram(alpha=0.5, bins=10) + \n",
    "        guides(fill=\"none\") +\n",
    "        facet_wrap(~ type, ncol=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question: do types 0, 1 and 2 have the same mean luminosity?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The t-test can only compare the means of two samples (or one sample with a theoretical mean). \n",
    "\n",
    "To move beyond two samples, we need to use a different method called [*analysis of variance*](https://en.wikipedia.org/wiki/Analysis_of_variance) (ANOVA)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### One-way ANOVA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Theory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0$: All of the groups have identical means:  $\\mu = \\mu_1 = \\mu_2 = \\mu_3$.\n",
    "\n",
    "$H_1$: Not all of the group means are identical.\n",
    "\n",
    "The one-way (also known as single-factor) ANOVA uses the F-test to compare the within-group and between-group variation:\n",
    "\n",
    "$$F = \\frac{\\text{between-group variation}}{\\text{within-group variation}}$$\n",
    "\n",
    "Under $H_0$, $F$ follows an F-distribution with parameters $(g-1,n_T-g)$, where $g$ is the number of groups (here, 3 types of star), and $n_T$ is the total number of observations.\n",
    "\n",
    "Once again, the F-distribution provides a p-value associated with the calculated value of $F$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Assumptions\n",
    "\n",
    "- Observations are independent.\n",
    "- Populations are normally distributed.\n",
    "- Variances of the populations are equal.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Application\n",
    "\n",
    "We will set $\\alpha=0.05$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "             Df Sum Sq Mean Sq F value   Pr(>F)    \n",
       "type          2  70.54   35.27   29.53 4.14e-11 ***\n",
       "Residuals   117 139.74    1.19                     \n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data %>%\n",
    "    filter(type %in% c(0,1,2)) %>%\n",
    "    aov(log(luminosity) ~ type, .) %>%\n",
    "    summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we have $p<\\alpha$, so we reject $H_0$: the three groups do not appear to have the same mean luminosity."
   ]
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    "### Other types of ANOVA"
   ]
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    "ANOVA is an important element of statistical analysis when we are interested in comparing the effects of different treatments. \n",
    "\n",
    "The underlying statistical model changes, depending on the expected relationship between treatment and effect (*fixed-*, *random-* or *mixed-effects*).\n",
    "\n",
    "Where multiple variables change simultaneously (for example, in patient populations), we may need to consider *multiple factors* (e.g. *two-way ANOVA*) and the *interactions* between factors.\n",
    "\n",
    "<br>"
   ]
  },
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   "metadata": {},
   "source": [
    "---"
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